Spinning probes and helices in AdS ₃

Abstract

We study extremal curves associated with a functional which is linear in the curve's torsion. The functional in question is known to capture the properties of entanglement entropy for two-dimensional conformal field theories with chiral anomalies and has potential applications in elucidating the equilibrium shape of elastic linear structures. We derive the equations that determine the shape of its extremal curves in general ambient spaces in terms of geometric quantities. We show that the solutions to these shape equations correspond to a three-dimensional version of Mathisson's helical motions for the centers of mass of spinning probes. Thereafter, we focus on the case of maximally symmetric spaces, where solutions correspond to cylindrical helices and find that the Lancret ratio of these equals the relative speed between the Mathisson-Pirani and the Tulczyjew-Dixon observers. Finally, we construct all possible helical motions in three-dimensional manifolds with constant negative curvature. In particular, we discover a rich space of helices in AdS 3 which we explore in

Entanglement, anomalies, and Mathisson’s helices

“…In this Section we study the general properties of extremal curves associated with the functional (1.5). The following discussion requires some acquaintance with extrinsic geometric terminology, we refer the reader to [12,13] for a detailed discussion and notation. To describe the geometry of a curve embedded in a three-manifold we must introduce a moving frame comprised by a normalized tangent vector t µ and two normal vectors n µ A , with A = 1, 2, defined by t µ n µ A = 0 , g µν n µ A n ν B = η AB , (2.1)…”

“…where D s = t µ ∇ µ is the directional derivative along the curve. In terms of these quantities, the equations that dictate the shape of the extremal curves can be written as [12]…”

“…The shape equations (2.3) can be solved analytically for maximally symmetric ambient manifolds. In [12] all possible solutions in AdS 3 spacetime,…”

“…We stress that the shape equations (2.3) are gauge covariant [12], hence the shape of the Mathisson helix itself is independent of the choice of frame. However, as mentioned before, this is not the case for the on-shell value of the functional (1.5).…”

“…(1.2). Difficulties arise from the fact that the extrema of (1.5) correspond to Mathisson's helical motions of spinning bodies [12] which are, in general, more complicated than geodesics. Moreover, since Mathisson's helices are solutions to higher order equations, it is necessary to fix more boundary conditions in order to single out a solution.…”